If you Google “exile” you will find that part of its definition is “to bar someone from their native country, typically for … punitive reasons.” In other words, a person who is exiled is forced to leave their country and cannot come back. They are away because they are being punished. This is not exactly how the term is used in Kipling’s poem “The White Man’s Burden.” It does refer to people being forced to leave their country, and it is a bad thing, but they are not really being punished. Instead, they are enduring punishing and negative conditions, but not because they are being punished for having broken the law.
In “The White Man’s Burden” Kipling emphasizes how hard it is for white people to go out and colonize other countries. He says that they have to go and “wait in heavy harness” on their subjects. He says that they have to work hard to help the natives. They have to build roads and ports they cannot use. They have to work hard while living and they have to “mark” the roads and ports “with your dead.”
They have to work hard to help the natives even though the natives will just ruin everything. The natives’ “sloth and heathen folly” will wreck all the things they are trying to achieve just as they are about to reach those goals. Moreover, the natives will resent them for all this. They will resent the whites for trying to civilize them.
What all this means is that it is very unpleasant to go out and be a member of an imperial administration. It is almost like being punished. The people who do this are being sent away from their native country. While abroad, they have to endure difficult conditions that are like what you would experience while being punished. This is what exile means in the context of this poem. It means being experiencing very unpleasant conditions while being away from your own home and country.